Finite-temperature topological entanglement entropy for fractons

Clare I. Abreu, Raúl Herrera, Edward H. Rezayi

Zhi Li, Roger S. K. Mong

We consider topological entanglement entropy (TEE) at finite temperature for CSS codes, which include some ordinary topological-ordered systems such as the toric code and some fracton models such as the Haah's code and the X-cube model. We find, under the assumption that there is no extended critical phase, the finite-temperature TEE is a piecewise constant function of the temperature, with possible discontinuities only at phase transitions. We then consider phase transitions of CSS codes. We claim that there must exist a phase transition at zero temperature for any CSS codes in 2D and 3D with topological order. This statement can be rigorous proved for some familiar examples, while for general models it can be argued based on the low-temperature expansion. This indicates the break down of topological orders at finite temperature. We also discuss possible connections with self-correcting quantum memory.

Bei Zeng

Han Ma, Albert T. Schmitz, S. A. Parameswaran +2 more

Entanglement entropy provides a powerful characterization of two-dimensional gapped topological phases of quantum matter, intimately tied to their description by topological quantum field theories (TQFTs). Fracton topological orders are three-dimensional gapped topologically ordered states of matter that lack a TQFT description. We show that three-dimensional fracton phases are nevertheless characterized, at least partially, by universal structure in the entanglement entropy of their ground-state wave functions. We explicitly compute the entanglement entropy for two archetypal fracton models, the ``X-cube model'' and ``Haah's code,'' and demonstrate the existence of a nonlocal contribution that scales linearly in subsystem size. We show via Schrieffer-Wolff transformations that this piece of the entanglement entropy of fracton models is robust against arbitrary local perturbations of the Hamiltonian. Finally, we argue that these results may be extended to characterize localization-protected fracton topological order in excited states of disordered fracton models.

Stefanos Papanikolaou, Kumar Raman, Eduardo Fradkin

Bowen Shi, Yuan-Ming Lu

The authors discuss a lower bound of nonlocal entanglement entropy (a generalization of topological entanglement entropy). The lower bound applies to Abelian conventional topological order as well as to type-I and type-II fracton models. It provides the physical intuition that each type of topological excitation, created by a deformable string, membrane, or fractal operator, gives a contribution to the nonlocal entanglement entropy. For fracton models, the lower bound is geometry dependent; it can take extensive values.